The author studies the map of a curve $z\left(t\right)=u\left(t\right)+iv\left(t\right)$ onto $w\left(t\right)=\int {z}^{2}\left(s\right)ds$ with appropriately chosen starting point for the integration. The latter curves are shown to have very convenient properties for differential-geometric computation since ${w}^{\text{'}}=({u}^{2}-{v}^{2})+2uvi$. If $u$, $v$ are polynomials of degree $n$, the components of $w$ are polynomials of degree $2n+1$. Special attention is given to Bézier control of the curves $w$.

The author complains about the lack of books using complex methods for real geometry. To his list of books should be added the classic “Inversive geometry” by *F. Morley* and *F. V. Morley* (1933; Zbl 0009.02908) and more recent books by *J. L. Kavanau* [e.g. Structural equation geometry (1983; Zbl 0541.51001) and Curves and symmetry (1982; Zbl 0491.51001)].